## Bounds

Bounds are the maximum and minimum values of a given number.

The rule to bounds is that the real value may be as much as half of the

**rounded unit**above or below the value.

In a bounds question, a value will be given and you will be asked to find either the upper or lower bound of that value. This is because the given value is already rounded, and you have to reverse it according to the question.

First, you must find the rounded unit in a question. For example, if a question says that

*'3.7m has been rounded to the nearest*then 0.1m is the rounded unit. This means that the real value is anything up to

**0.1m**'**3.7m ± 0.05m**. The upper bound is always adding the 0.05 as it is the higher possibility, resulting in the answer being

**3.75**. The lower bound is

subtracting the 0.05 as it is the lower possibility, resulting in the answer

**3.65**.

An example of an actual bounds question appearing in an IGCSE paper is:

*"There are 1300 sheets of paper, correct to the nearest 100 sheets, in a pile. Each sheet is of equal thickness.*

*The height of the pile is 160 mm, correct to the nearest 10mm.*

*Calculate the upper bound in millimetres, for the thickness of one sheet of paper."*

(From Mathematics A, paper 4H, Wednesday 15 January 2014 - Morning)

In this question we have to find the highest value for the thickness of one sheet of paper as it asks for the

**upper bound**. to find the thickness of one sheet of paper in this question, we would need to do

**height ÷ sheets**. (This is where it gets tricky). To find the highest value from this equation, we would need to do

**Upper bound ÷ lower bound**. (because a higher value divided by a lower value gives a higher answer than any other combination).

To find the upper bound of the Height:

*"The height of the pile is 160 mm, correct to the*

**nearest 10mm**"The rounded unit is 10mm, so the answer would be 160mm ± 5mm. As it is the upper bound, we add 5mm, meaning the upper bound of the sheets is

**165mm**.

To find the lower bound of the number of sheets:

*"There are 1300 sheets of paper, correct to the*

**nearest 100 sheets**"The rounded unit is 100 sheets, so the answer would be 1300 ± 50 sheets. As it is the lower bound, we subtract50 sheets, meaning the lower bound of the number of sheets is

**1250**.

We then complete the

**height ÷ sheets**, which would be

**165 ÷ 1250**, giving the answer of

**0.132mm**as the upper bound of the thickness of one sheet of paper.

**Maximum and minimum values**

When a calculation is done, sometimes it is done using rounded values. This will mean that the answer will not be accurate because the values used in the calculation were rounded and not accurate. For this, you may be asked to find maximum and minimum values. An example is:

*"The area of a garden is measured to be 6.2m x 3.3m to the nearest 10cm. Find the maximum and minimum values for the area of the garden."*

The area of the garden using the rounded off values gives an answer of

*20.46m*

*²*

*. The maximum value of this equation would be the highest value it could possibly be. Since it was rounded to the nearest 10cm, the highest values would be +5cm (0.05m) and the lowest values would be -5cm (0.05m). To find the maximum value, we would have to add 5cm to both parts of the equation and to find the minimum value, we would have to subtract 5cm from both sides of the equation. So:*

- Maximum value =

*6.25m x 3.35m = 20.9375m*

*²*

- Minimum value =

*6.15m x 3.25m = 19.9875m*

*²*